Genomic length DNA molecules have served as a model system for studying the physics of single polymers, in part due to their large lengthscales, compatibility with fluorescence optical microscopy, excellent monodispersity, and the commercial availability of complete genomes of the lambda and T4 viruses. There has been theoretical interest in the physics of two dimensional polymers, but robust experimental systems have been lacking. Here, we extend the use of DNA as a model polymer into the second dimension by studying the physics of the kinetoplast in free solution. A kinetoplast is a complex genomic structure found in certain parasites that consists of thousands of topologically linked rings of DNA forming a two-dimensional network. Removed from the cell and viewed in fluorescence microscopy in good solvent conditions, it adopts a form akin to a jellyfish bell approximately 5 microns in diameter. We characterize the equilibrium conformation and dynamics of kinetoplasts and examine their response to flow, confinement, and varying solvent conditions.